Lobachevskii Journal of Mathematics

http://ljm.ksu.ru

Vol. 18, 2005, 33–51 \begin{matrix}c \\ ◯ \end{matrix} Mohammed Benalili, Azzedine Lansari

Mohammed Benalili and Azzedine Lansari

SPECTRAL PROPERTIES OF THE ADJOINT OPERATOR AND APPLICATIONS

(submitted by M.A. Malakhaltsev)

Abstract. We present some spectral properties of the adjoint operator corresponding to an admissible dilatation vector field and its perturbations. Next, we apply these results via the Nash-Moser function inverse theorem to show that the group G of diffeomorphisms on the Euclidean space R^{n} which are 1-time flat, close to the identity and of small support acts transitively on the affine space of appropriate perturbations of the dilation vector field X_{o}.

1. Introduction

Let R^{n} be the euclidean space endowed with the norm ∥.∥ and 𝟋 be the Schwartz space of all functions on R^{n} which are fast falling together with all derivatives. The convergence is defined by the seminorms

∥f∥_{k} = max_{m+|α|≤k} sup_{x∈R^{n}} \{(1 + ∥x∥^{2})^{\frac{m}{2}} ∥D^{α}f(x)∥\}

where k ∈ N and α = (α_{1}, ..., α_{n}) ∈ N^{n} is a multi-indices with length |α| = α_{1} + ... + α_{n} and the sup runs over all partial derivatives D^{α} of degree |α| at most k. Denote by j_{0}^{1}F the closed subspace of 𝟋 of 1-time flat functions at the origin 0 with the induced gradings.

Knowing from the fundamental work of A.Zajtz [5] that the right invertibility of the derivative of the orbital projection and that of the exponential map reduce to the invertibility of the adjoint operator, we give in the first part of this work some spectral properties related to the adjoint operator corresponding to an admissible dilatation vector field and some of its perturbations.

In the second part of our paper we give some applications of the study done in the first part of this work via the Nash-Moser theorem, essentially we establish that the group G of diffeomorphisms f of R^{n} close to the identity and with small support such that f −id ∈ (j^{1}F)^{n}( which is tame Lie Fréchet group) acts transitively on the affine space of the perturbed vector fields of the admissible dilatation vector field X_{o}. As a second application, we state that any diffeomorphism from the shifts Go exp(X_{o}) or exp(X_{o})oG in a neighborhood of exp(X_{o}) are the value at time1 of a smooth flow. This latter result gives an affirmative answer to the question: what diffeomorphism on a manifold is imbeddible in a smooth flow in contrast with many negative answer on compact manifold.

We quote as examples:

Negative results:

(N. Kopell 1970) Let D^{∞}([0, 1]) denote the group of smooth diffeomorphisms f of [0, 1] such that f − id has all derivatives globally bounded. There are f ∈ D^{∞}([0, 1]) arbitrary close to the identity in the C^{∞} topology such that f does not embed in a C^{1}- flow and has no fixed point.

(M.I. Bryn 1974) The subset of all C^{2} diffeomorphisms of a smooth compact manifold which embed in a C^{1} -flow is nowhere dense in the set of all Morse-Smale diffeomorphisms.

(A. Zajtz 1996) No Anosov diffeomorphism embeds into a flow.

Positive results:

(S. Steinberg 1958) Let f be a local C^{∞} volume preserving diffeomorphism of R^{n}, n ≥ 2, keeping the origin fixed, with eigenvalues satisfying if λ_{i} = λ_{1}^{m_{1}}...λ_{n}^{m_{n}} then m_{i} − 1 = m_{j} for all j ≠ i, lies in one parameter group.

(J. Palis, S. Palais 1969) in D^{1} there exists an open set of Morse-Smale diffeomorphisms imbeddible in a topological flow.

(J.Grabovski 1988) Every orientation preserving homeomorphism of the interval [0, 1] embeds in a topological flow.

2. Spectral properties of the adjoint operators

Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be a linear vector field on R^{n}( the dilatation vector field), we assume that:

(i) all the coefficients α_{i} are nonzero and of the same sign

(ii) for any multi-indices m = (m_{1}, ..., m_{n}) ∈ N^{n} with length |m| = m_{1} + ... + m_{n} ≥ 2, and

a = min {|α_{i}| : i = 1, ..., n} , b = max {|α_{i}| i = 1, ..., n}

we have |m| a − b ≥ ρ > 0.

SPECTRAL PROPERTIES OF THE ADJOINT OPERATOR 35

X_{o} satisfying the conditions (i) and (ii) will be called admissible infinitesimal dilatation. X_{o} induces a global flow of dilatations (ϕ_{t}) given by

ϕ_{t}(x) = exp (tX_{o}) (x) = (x_{1} exp α_{1}t, ..., x_{n} exp α_{n}t).

Let Υ_{1}(t) = (ϕ_{t})_{∗} and Υ^{1}(t) = (ϕ_{t})^{∗}be the adjoint diffeomorphisms associated to the infinitesimal generators ad(−X_{o}) and ad(X_{o}) defined by

(ϕ_{t})_{∗}Y (x) = (Dϕ_{t}.Y)oϕ_{−t}(x)

and

(ϕ_{t})^{∗}Y (x) = (Dϕ_{−t}.Y)oϕ_{t}(x).

Denote by 𝟋 the graded Fréchet space of C^{∞}−functions on R^{n} with gradings given in the introduction. Let j_{0}^{1}𝟋 denote the closed subspace of 𝟋 of functions f which are 1-time flat at the origin 0 that is to say f(0) = Df(0) = 0, with the induced gradings. Denote by j_{0}^{1}E the graded Fréchet space of vector fields defined on R^{n} and with components in j_{0}^{1}𝟋.

For any vector field Y = ∑_{i=1}^{n} u_{i}(x)\frac{∂}{∂x_{i}} , we have

Υ_{1}(t)Y = (ϕ_{t})_{∗}Y = ∑_{j=1}^{n} e^{α_{j}t}u_{j}(ϕ_{−t}(x))\frac{∂}{∂x_{j}}

and

Υ^{1}(t)Y = (ϕ_{t})^{∗}Y = ∑_{j=1}^{n} e^{−α_{j}t}u_{j}(ϕ_{t}(x))\frac{∂}{∂x_{j}} .

Then, for any multi-indices ζ = (ζ_{1}, ..., ζ_{n}) ∈ N^{n} and any unit vector v = (v_{1}, ..., v_{n}) ∈ R^{n}, we put v^{ζ} = (v_{1}^{ζ_{1}}, ..., v_{n}^{ζ_{n}}) .

By derivation, we get

D_{x}^{ζ}Υ_{1}(t)Y.v^{ζ} = ∑_{i=1}^{n} e^{α_{i}t}D_{y}^{ζ} u_{i}(y) (D_{x}ϕ_{−t}(x)v)^{ζ} \frac{∂}{∂x_{i}}

= ∑_{i=1}^{n} exp t(α_{i} − ∑_{j=1}^{n} ζ_{j}α_{j})D_{y}^{ζ} u_{i}(y)v^{ζ} \frac{∂}{∂x_{i}} ,

where y = ϕ_{t}(x).

2.1. Spectrum of the operator λI − Υ(t). First, let K ⊂ R^{n} be a compact neighborhood of the origin 0 ∈ R^{n}. Then we establish

Lemma 1. Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be an admissible infinitesimal dilatation and Y = ∑_{i=1}^{n} u_{i}(x)\frac{∂}{∂x_{i}} be any vector field in the Fréchet space j_{0}^{1}E and Υ_{1}(t) denotes (ϕ_{t})_{∗} (resp. Υ(t) denotes (ϕ_{t})^{∗}). Then

(i) If all α_{i} are positive (α_{i} > 0), the series ∑_{m≥0} (Υ(t))^{m} Y converges uniformly in x ∈ K with respect to t ∈ R^{∗+} on the Fréchet space j_{0}^{1}E .

(ii) In case all α_{i} are negative (α_{i} < 0), the series ∑_{m≥0} (Υ^{1}(t))^{m} Y converges uniformly in x ∈ K with respect to t ∈ R^{∗+} on j_{0}^{1}E.

Proof. ( i) Let K ⊂ R^{n} be a compact neighborhood of the origin 0 ∈ R^{n} and D = K × R^{∗+}. For any (x, t) ∈ D and any vector field Y = ∑_{i=1}^{n} u_{i}(x)\frac{∂}{∂x_{i}} ∈ j^{1}E, we consider the series

∑_{m≥0} (ϕ_{t∗})^{m} Y = ∑_{m≥0} (ϕ_{mt ∗}) Y = ∑_{j=1}^{n} (∑_{m≥0} e^{mtα_{j}}u_{j}(ϕ_{−mt}(x))) \frac{∂}{∂x_{j}} .

For any integer r ≥ 0 and any multi-indices β = (β_{1}, ..., β_{n}) ∈ N^{n} with length |β| = β_{1}+ ... + β_{n}, we obtain by derivation

∥D_{x}^{β} (Υ_{1}(t))^{m} Y ∥_{r} = ∥D_{x}^{β} (ϕ_{t∗})^{m} Y ∥_{r}

= max_{i=1,...,n} sup_{x∈R^{n}} \{(1 + ∥x∥^{2})^{\frac{m}{2}} |D_{x}^{ζ+β}u_{i}(ϕ_{−mt}(x))|

m + |ζ + β| ≤ r + |β|, x ∈ K}

Putting ξ = ζ + β, y = ϕ_{−mt}(x) , we get

D_{x}^{ξ}u_{i}(ϕ_{−mt}(x)) = exp(−mt ∑_{i=1}^{n} α_{i}ξ_{i})D^{ξ}u_{i}(y)

so

∥D_{x}^{β} (Υ_{1}(t))^{m} Y ∥_{r}

≤ max_{i=1,...,n} sup \{(1 + ∥x∥^{2})^{\frac{m}{2}} |D_{y}^{ξ}u_{i}(ϕ_{−mt}(x))| : m + |ξ| ≤ r + |β| , x ∈ R^{n}\}

× exp (−mt(∑_{j=1}^{n} ξ_{j}α_{j} − α_{i}))

≤ ∥Y ∥_{r+|β|} exp (−mt(∑^{n} ; j=1ξ_{j}α_{j} − α_{i})) .

So if |ξ| ≥ 2, by the admissibility of the infinitesimal dilatation X_{o},we get

∥D_{x}^{β} (Υ_{1}(t))^{m} Y ∥_{r} ≤ ∥Y ∥_{r+|β|} exp(−ρmt)

and

∑_{m≥0} ∥D_{x}^{β} (Υ_{1}(t))^{m} Y ∥_{r} ≤ ∥Y ∥_{r+|β|} \frac{1}{1 − e^{−ρt}}  (1)

for any (x, t) ∈ K × R^{∗+}.

In the case |ξ| ≤ 1, we use the 1-time flatness of the vector field Y: that is there is a constant M > 0 such that |u_{i}(x)| ≤ ∥x∥^{2} and |Du_{i}(x)| ≤ M ∥x∥ for any x ∈ K and i = 1, ..., n. Then

|e^{α_{i}mt}D_{x}^{ξ}u_{i}(ϕ_{−mt}(x))| ≤ Me^{(α_{i}−2α_{o})mt}

where α_{o} = inf {α_{i} : i = 1, ..., n} . And by the assumption on the infinitesimal dilatation, we get the estimation 1.

(ii) It remains to check that the time depending vector field X(t) = (ϕ_{t})_{∗} Y belongs to the Fréchet space j_{0}^{1}E . It is obvious that if the vector field Y is 1-flat so do the vector field X and the same computations as part (i) of the proof able us to conclude. □

As a Corollary of Lemma1, we have

Lemma 2. Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be an admissible infinitesimal dilatation. Then for any t ∈ R^{∗+}, the linear operator (I − ϕ_{t∗}) with α_{i} > 0, i = 1, ..., n, (resp. (I − ϕ_{t}^{∗}) in case α_{i} < 0, i = 1, ..., n) is invertible in the Fréchet space j_{0}^{1}E, and its inverse is given by

(I − ϕ_{t∗})^{−1}Y = ∑_{m≥0} (ϕ_{t∗})^{m} Y = ∑_{i=1}^{n} (∑_{m≥1} e^{mtα_{i}}u_{i}(ϕ_{−mt}(x))) \frac{∂}{∂x_{i}}

(resp. (I − ϕ_{t}^{∗})^{−1}Y = ∑_{m≥0} (ϕ_{t}^{∗})^{m} Y = ∑_{i=1}^{n} (∑_{m≥0} e^{−mtα_{i}}u_{i}(ϕ_{mt}(x))) \frac{∂}{∂x_{i}})

Let C be the complex field for any t > 0 and ρ > 0 the real number of the condition (ii) of the admissibility of the dilatation X_{o} that is to say: for any multi-indices m = (m_{1}, ..., m_{n}) ∈ N^{n} with length |m| the constants a = min {|α_{i}| i = 1, ..., n} and b = max {|α_{i}| i = 1, ..., n} fulfill the relation |m| a − b ≥ ρ > 0. Letting ρ(Υ(t)) = {λ ∈ C : |λ| > e^{−ρt}} and K be any compact neighborhood of the origin 0 ∈ R^{n}, we have

Theorem 1. Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be an admissible infinitesimal dilatation. For any λ ∈ ρ(Υ(t)) and any (x, t) ∈ D = K ×R ^{∗+}, the linear operator (λI − ϕ_{t ∗}) with α_{i} > 0, i = 1, ..., n (resp. (λI − ϕ_{t}^{∗}) in case α_{i} < 0 ) is invertible on the Fréchet space j_{0}^{1}E. The inverse operator on j_{0}^{1}E is given by R(λ, Υ(t))Y = (λI − Υ(t))^{−1}Y = ∑_{m≥0} \frac{1}{λ^{m+1}} (Υ(t))^{m} Y where Υ(t) denotes ϕ_{t ∗} (resp. ϕ_{t}^{∗}).

Proof. Let R(λ, Υ_{1}(t)) denotes the resolvent of the adjoint operator Υ_{1}(t) = ϕ_{t∗}, we have

R(λ, Υ_{1}(t))Y = (λI − ϕ_{t∗})^{−1}Y = λ^{−1}(I − \frac{1}{λ}ϕ_{t∗})^{−1}Y

= ∑_{m≥0} \frac{1}{λ^{m+1}} (ϕ_{t∗})^{m} Y = ∑_{i=1}^{n} (∑_{m≥0} \frac{1}{λ^{m+1}} e^{mα_{i}t}u_{i}(ϕ_{−mt}(x))) \frac{∂}{∂x_{i}} .

The rest of the proof is similar to that of Lemma1. □

Corollary 1. For every t > 0, the spectrum σ (Υ(t)) of the adjoint operator Υ(t) = (ϕ_{t})_{∗} defined on the Fréchet space j^{1}E (resp. Υ(t) = (ϕ_{t})^{∗}) is contained in the closed ball centered at the origin and of radius e^{−ρt}.

2.2. Right invertibility of the differential operator λI − ad(X_{o}). Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be an admissible infinitesimal dilatation and j_{0}^{1}E be the graded Fréchet space defined in the previous section.

Theorem 2. If all the coefficients are positive (α_{i} > 0), then for any complex λ with nonpositive real part, the differential operator λI −ad(X_{o}) is surjective on the Fréchet space j_{0}^{1}E. In the case where all α_{i} are negative (α_{i} < 0) and the complex λ with nonnegative real part, the differential operator λI − ad(X_{o}) is surjective on j_{0}^{1}E.

Proof. Letting Y = ∑_{i=1}^{n} f_{i}(x)\frac{∂}{∂x_{i}} ∈ j_{0}^{1}E, we look for a vector field Z = ∑_{i=1}^{n} u_{i}(x)\frac{∂}{∂x_{i}} in the Fréchet space j_{0}^{1}E such that

Y = (λI − ad(X_{o})) Z = λZ − [X_{0}, Z] .  (2)

1) Consider the case α_{i} < 0, i = 1, ..., n and the real part of the complex λ, Re(λ) ≥ 0 , and let a and b be real constants fulfilling 0 <a ≤ −α_{i} ≤ b.

We claim that a solution of the equation2 is given

Z = ∫_{0}^{+∞} e^{−λt} (ϕ_{t})^{∗} Y dt  (3)

In fact, equation(2.3) writes in coordinates

− ∑_{j=1}^{n}(α_{j}x_{j})\frac{∂u_{i}(x)}{∂x_{j}} + (λ + α_{i})u_{i}(x) = f_{i}(x)  (4)

and we have to check that the function

u_{i}(x) = ∫_{0}^{+∞} e^{−t(λ+α_{i})}f_{i}(e^{tα_{1}}x_{1}, ..., e^{tα_{n}}x_{n})dt  (5)

is well defined and constitutes a solution of equation 4. We do it in the following steps

(i) First we show that the integral u_{i} converges uniformly. Letting r be nonnegative integer and ν = (ν_{1}, ..., ν_{n}) any multi-indices with length |ν| = ν_{1} + ... + ν_{n} ≤ r, we get by taking the derivative

D_{x}^{ν}e^{−t(λ+α_{i})}f_{i}(e^{tα_{1}}x_{1}, …, e^{tα_{n}}x_{n}) = exp t(−α_{i} − λ + ∑_{j=1}^{n} ν_{j}α_{j})D_{y}^{ν}f_{i}(y),

where y_{i} = e^{α_{i}t}x_{i}.

Then ∫_{0}^{+∞} |D_{x}^{υ} (e^{−t(λ+α_{i})}f_{i}(e^{tα_{1}}x_{1}, ..., e^{tα_{n}}x_{n}))| dt

≤ ∥Y ∥_{r} ∫_{0}^{+∞} exp t(−α_{i} − Re(λ) + ∑_{j=1}^{n} ν_{j}α_{j})dt

≤ ∥Y ∥_{r} ∫_{0}^{+∞} exp t(−Re(λ) + b − |ν| a)dt.

So if |ν| ≥ 2, by the admissibility of the infinitesimal dilatation we get

∫_{0}^{+∞} exp t(−Re(λ) + b − |ν| a)dt ≤ ∫_{0}^{+∞} exp t(−Re(λ) − ρ)dt

= \frac{1}{ρ + Re(λ)}

and the integral converges uniformly.

In the case |ν| ≤ 1, we use the flatness of the vector field Y at the origin 0.

(ii) We have to verify that the vector field Z = ∑_{i=1}^{n} u_{i}(x)\frac{∂}{∂x_{i}} belongs to the graded Fréchet space j_{0}^{1}E, but this will be easily checked by the same calculation as in step(i).

(iii) It remains to show that u_{i} is a solution of equation 4. By direct computations we get

∑_{j=1}^{n} α_{j}x_{j} \frac{∂u_{i}(x)}{∂x_{j}} = ∫_{0}^{+∞} e^{−t(λ+α_{i})} \frac{d}{dt}f_{i}(e^{tα_{1}}x_{1}, ..., e^{tα_{n}}x_{n})dt

= [e^{−t(λ+α_{i})}f_{i}(e^{tα_{1}}x_{1}, ..., e^{tα_{n}}x_{n})]_{0}^{+∞} + (λ + α_{i})u_{i}(x)

so

∑_{j=1}^{n} α_{j}x_{j} \frac{∂u_{i}(x)}{∂x_{j}} = −f_{i}(x) + (λ + α_{i})u_{i}(x).

2) In the case all α_{i} > 0, a solution of 4 is given by

Z = ∫_{0}^{+∞} e^{λt} (ϕ_{t})_{∗} Y dt

with the real part of the complex λ, Re(λ) ≤ 0. □

Remark 1. By doing the change of coordinates t = e^{−τ}, 5 writes

u_{i}(x) = − ∫_{0}^{1} t^{−1+λ+α_{i}}f_{i}(t^{α_{1}}x_{1}, ..., t^{α_{n}}x_{n})dt.

2.3. Surjectivity of the perturbed adjoint operator. Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be the infinitesimal dilatation on R^{n} and Y_{o} = X_{o} + Z a perturbed vector field of X_{o}.

Assume that

(i) all the coefficients α_{i} < 0 (ii) the perturbation Z satisfies the conditions

∥D^{l}Z(x)∥ ≤ \{ \begin{matrix}\{ \begin{matrix}c_{l} ∥x∥^{k} & if ∥x∥ < 1 \\ c_{l} & if ∥x∥ ≥ 1 \end{matrix} & for l = 0, 1 and any integer k ≥ 1 \\ \{ \begin{matrix}c_{l} ∥x∥^{k−l} & if ∥x∥ < 1 \\ c_{l} & if ∥x∥ ≥ 1 \end{matrix} & for 2 ≤ l < k \\ c_{l} & for any x ∈ R^{n}. \end{matrix}  (P)

where c_{l} is a constant.

Letting a = min_{i=1,...,n}(|α_{i}|), b = max_{i=1,...,n}(|α_{i}|),put

a_{l} = a − c_{l} > 0 for l = 0, 1.
a_{l} = c_{l}  l ≥ 2
b_{l} = b + c_{l}
a′_{1} = min {a_{o}, a_{1}}
b′_{1} = min {b_{o}, b_{1}} .

With the above notations, the perturbed vector field Y_{o} = X_{o} +Z will be said admissible if

(i) all the coefficients α_{i} are nonzero and of the same sign (ii) For any multi-indices m = (m_{1}, ..., m_{n}) ∈ N^{n} with length |m| ≥ 2, |m| a′_{1} − b′_{1} ≥ ρ > 0 where ρ is a positive number. (iii) the perturbation Z satisfies the condition (P). Let ψ_{t} be the flow induced by the vector field Y_{o} = X_{o} + Z. The second author stated in [4] a result of global stability. Before

quoting this result, we remind :

Definition 1. An equilibrium point a ∈ R^{n} of a flow (ϕ_{t}) is said globally asymptotically stable if

(i) a is an asymptotically stable equilibrium of the flow (ϕ_{t})

(ii) For any compact set K ⊂ R^{n} and any ε > 0 there exists T_{K} > 0 such that for each t ≥ T_{K} we have ∥ϕ_{t}(x) − a∥ ≤ ε for all x ∈ K.

Lemma 3. ([4]) Let K be a compact neighborhood of the origin 0 ∈ R^{n}. Suppose that all coefficients of the dilatation X_{o} are α_{i} < 0 and the perturbed vector field Y_{o} = X_{o}+ Z is admissible. Then the origin 0 is globally asymptotically stable and (under the above notations) there are constants M_{l} > 0 such that

(a) ∥x∥ e^{−b_{o}t} ≤ ∥ψ_{t}(x)∥ ≤ ∥x∥ e^{−a_{o}t} for any t > 0.
(b) M_{1}e^{a′_{1}t} ≤ ∥Dψ_{−t}(x)∥ ≤ M_{1}e^{b′_{1}t}  for any (x,t) ∈ K × R^{∗+}
∥D^{l}ψ_{t}(x)∥ ≤ M_{l}e^{−a′_{1}t}  for any (x,t) ∈ K × R^{∗+}

and any integer l ≥ 2.

Having Lemma3 in mind, we state

Theorem 3. Assume that the perturbed vector field Y_{o} = X_{o} + Z is admissible , then for any complex λ with nonpositive (resp. nonnegative) real part the differential operator λI − ad(Y_{o}) is surjective on the Fréchet space j_{0}^{1}E provided that all the coefficients α_{i} of the linear part X_{o} of Y_{o} are positive (resp. all α_{i} < 0).

Proof. Let Y = ∑_{i=1}^{n} f_{i}(x)\frac{∂}{∂x_{i}} ∈ j_{0}^{1}E, we look for a vector field X = ∑_{i=1}^{n} u_{i}(x)\frac{∂}{∂x_{i}} from the Fréchet space j_{0}^{1}E such that

Y = (λI − ad(Y_{o})) X = λX − [Y_{o}, X] .  (6)

If the integral

X = ∫_{0}^{+∞} e^{λt} (ψ_{t})_{∗} Y dt (resp. X = ∫_{0}^{+∞} e^{−λt} (ψ_{t})^{∗} Y dt )  (7)

converges uniformly in the Fréchet space j_{0}^{1}E , then the operator λI − ad(Y) is right invertible in j_{0}^{1}E and the equation (6) admits as a solution the vector field given by (7).

We have to show that the integral (7) converges.

(i) Suppose all the α_{i} < 0 and the real part of the complex λ, Re(λ) ≥ 0. Let r ∈ N and η = (η_{1}, ..., η_{n}) ∈ N^{n} be any multi-indices with length |η| = η_{1} + ... + η_{n} ≤ r ; we show that the following integral

I = ∫_{0}^{+∞} e^{−Re(λ)t}D_{x}^{υ} (Dψ_{−t}(ψ_{t}(x)).Y ∘ ψ_{t}(x)) dt

converges uniformly for any Y ∈ j_{0}^{1}E. □

In coordinates, we have

D_{x}^{η} (Dψ_{−t}(ψ_{t}(x)).Y ∘ ψ_{t}(x))_{i} = D_{x}^{η} (∑_{j=1}^{n} \frac{∂}{∂x_{j}} ψ_{i}(−t, ψ_{t}(x)).f_{j} ∘ ψ_{t}(x)) .

Letting, for any multi-indices k = (k_{1}, .., k_{n}) ∈ N^{n},

D_{x}^{k}ψ_{−t}(x) = \frac{∂^{|k|}}{∂x_{1}^{k_{1}}...∂x_{n}^{k_{n}}} ψ_{−t}(x) = ∏_{i=1}^{n} \frac{∂^{k_{i}}}{∂x_{i}^{k_{i}}} ψ_{−t}(x)

we get

D_{x}^{η} (Dψ_{−t}(ψ_{t}(x)).Y ∘ ψ_{t}(x))_{i} = ∑_{j=1}^{n} ∏_{ι=1}^{n} \frac{∂^{η_{ι}}}{∂x_{ι}^{η_{ι}}} (\frac{∂ψ_{i}}{∂x_{j}} (−t, ψ_{t}(x)).f_{j} ∘ ψ_{t}(x))

= ∑_{j=1}^{n} ∏_{ι=1}^{n} ∑_{k_{ι=0}}^{η_{ι}} C_{η_{ι}}^{k_{ι}} \frac{∂^{η_{ι}−k_{ι}}}{∂x_{ι}^{η_{ι}−k_{ι}}∂x_{j}} ψ_{i} (−t, ψ_{t}(x)) .\frac{∂^{k_{ι}}}{∂x_{ι}^{k_{ι}}} f_{j} ∘ ψ_{t}(x)

= ∑_{j=1}^{n} ∑_{k_{1}=0,...,η_{1}_{.......................} ; k_{n}=0,...,η_{n}} C_{η}^{k}D_{x}^{η−k}ψ_{i} (−t, ψ_{t}(x)) .D_{x}^{k}f_{j} ∘ ψ_{t}(x)

where

C_{η}^{k} = \frac{η!}{k!(η − k)!} and k! = k_{1}!...k_{n}!.

Putting ν = η − k i.e. ν_{i} = υ_{i} − k_{i} and letting v = (v_{1}, ..., v_{n}) ∈ R^{n} be any unit vector, we obtain

D_{x}^{υ−k}ψ_{i} (−t, ψ_{t}(x)) v^{ν} = D_{x}^{ν}ψ_{i}(−t, y)v^{ν}

= ∑_{l_{1}=1}^{ν_{1}} ... ∑_{l_{n}=1}^{ν_{n}} D_{y}^{l}ψ_{i}(−t, y) ∏_{i=1}^{n} ∑_{ζ_{1}^{i}+...+ζ_{l_{i}}^{i}=ν_{i}}D_{x}^{ζ_{1}^{i}}ψ_{t}(x)v^{ζ_{1}^{i}}...D_{x}^{ζ_{l_{i}}^{i}}ψ_{t}(x)v^{ζ_{l_{i}}^{i}}

and

D_{x}^{k}f_{j} ∘ ψ_{t}(x) = D_{x}^{k}f_{j}(y)

= ∑_{l_{1}=1}^{k_{1}} ... ∑_{l_{n}=1}^{k_{n}} D_{y}^{l}f_{j}(y) ∏_{i=1}^{n} ∑_{ζ_{1}^{i}+...+ζ_{l_{i}}^{i}=k_{i}}D_{x}^{ζ_{1}^{i}}ψ_{t}(x)v^{ζ_{1}^{i}}...D_{x}^{ζ_{l_{i}}^{i}}ψ_{t}(x)v^{ζ_{l_{i}}^{i}}.

By Lemma3 there are positive constants M_{l} > 0 fulfilling for any (x, t) ∈ K × R^{∗+}

M_{1}e^{a′_{1}t} ≤ ∥D ψ_{−t}(x)∥ ≤ M_{1}e^{b′_{1}t}
∥D^{l}ψ_{t}(x)∥ ≤ M_{l}e^{−a′_{1}t} with l ≥ 2

so there exist constants C_{1} > 0 and C_{2} > 0 such that

∥D_{x}^{υ−k}ψ_{i} (−t, ψ_{t}(x)) v^{ν}∥ ≤ C_{1} exp (−(|υ| − |k|)a′_{1} + b′_{1}) t

and

∥D_{x}^{k}f_{j} ∘ ψ_{t}(x)∥ ≤ C_{2} ∥D_{y}^{k}f_{j}(y)∥ exp(− |k| a′_{1}t).

Consequently

I ≤ max_{1≤i≤n} ∫_{0}^{+∞} e^{−Re(λ)t} |D_{x}^{η} (Dψ_{−t}(ψ_{t}(x)).Y ∘ ψ_{t}(x))_{i}| dt

≤ ∥Y ∥_{r} ∫_{0}^{+∞} e^{−t(Re(λ)+|υ|a′_{1}−b′_{1})}dt

so if |η| ≥ 2, by the admissibility of the perturbed vector field Y_{o}, we get

I ≤ ∥Y ∥_{r} ∫_{0}^{+∞} e^{−t(Re(λ)+ρ)}dt = \frac{1}{ρ + Re(λ)}.

In the case |η| ≤ 1, we use the flatness of the vector field Y at the origin to conclude.

To check that

X = ∫_{0}^{+∞} e^{−λt} (ψ_{t})^{∗} Y dt

is a solution of the Equation 6, it suffices to remark that

[Y_{o}, X] = \frac{d}{ds} |_{s=0} (ψ_{s})^{∗}X = \frac{d}{ds} |_{s=0} (ψ_{s})^{∗} ∫_{0}^{+∞} e^{−λt} (ψ_{t})^{∗} Y dt

= \frac{d}{ds} |_{s=0} ∫_{0}^{+∞} e^{−λt} (ψ_{t+s})^{∗} Y dt

= ∫_{0}^{+∞} e^{−λt} \frac{d}{dt} (ψ_{t})^{∗} Y dt

= e^{−λt} (ψ_{t})^{∗} Y |_{0}^{∞} +λ ∫_{0}^{+∞} e^{−λt} (ψ_{t})^{∗} Y dt
= −Y + λX.

(ii) In the case all α_{i} > 0, and Re(λ) ≤ 0, the arguments are as in part (i) .

3. Applications to dynamic systems and to the exponential map

3.1. Category of tame Fréchet manifolds.

Definition 2. Let E and F be Fréchet spaces with graded countable collections of seminorms. We say that a non linear map P : U ⊂ E → F satisfies a tame estimate of degree r and base b if

∥P (f)∥_{n} ≤ C(1 + ∥f∥_{n+r})

for all f ∈ U and all n ≥ b, and a constant C which may depend on n, P is a tame map if it is defined on an open set, is continuous, and satisfies a tame estimate in a neighborhood of each point.

3.2. General estimates. Applying the product and the chain rules of the differentiation and using the interpolation formulae (see [3]) one gets the estimates on any compact set K of R^{n}.

∥fg∥_{k} ≤ C (∥f∥_{k} ∥g∥_{0} + ∥f∥_{0} ∥g∥_{k})  (8)
∥D^{k} (f ∘ g)∥ ≤ C ∥g∥_{1}^{k−1} (∥f∥_{k} ∥g∥_{1} + ∥f∥_{1} ∥g∥_{k})  (9)
∥f^{−1}∥_{k} ≤ C_{k} (1 + ∥f∥_{k})  (10)

for all k ≥ 1 the latter estimate holds if

∥f − id∥ ≤ ε where ε is small.

Example 1. (of tame map) If f is a diffeomorphism on a compact manifold M close to the identity then the adjoint operators f_{∗} = Ad(f) and its inverse f^{∗} = Ad(f^{−1}) are tame of degree 0 and base 0 on the space of vector fields on M

In fact from the general above estimates, we get, for any vector field on M,

∥f_{∗}X ∥_{n} = ∥(Df.X)of^{−1}∥_{n}
≤ ∥f^{−1}∥_{1}^{n−1} (∥Df.X∥_{n} ∥f^{−1}∥_{1} + ∥Df.X∥_{1} ∥f^{−1}∥_{n})
≤ C_{1} (1 + ∥f∥_{n})^{n} ∥f∥_{n+1} ∥X∥_{n}

so

∥f_{∗}X ∥_{n} ≤ C ∥X∥_{n}

where C is a constant.

Example 2. In particular, any continuous map from E to a Banach space and any continuous map of finite dimension space into F are tame.

Definition 3. P is a smooth tame map if it is smooth and all its derivatives are tame. A linear map L : E → F is tame if it satisfies a tame estimate

∥L(f)∥_{n} ≤ C ∥f∥_{n+r}

for some r and all n ≥ b.

Definition 4. A graded Fréchet space E is tame if it is direct summand of a space ∑(B) of exponentially decreasing sequences in a Banach space B, so that we have

E →^{L} ∑(B) →^{M} E

with MoL = id_{E} and M, L are tame.

Example 3. By a tame manifold we mean a smooth manifold modeled on a tame Fréchet space; in this category we have also the notion of tame Lie group (see [3]).

We quote the Nash-Moser function theorem

Theorem 4. (Nash, Moser) Let F and G be tame space and P : F → G a smooth tame map. Suppose that the equation for the derivative DP (f)h = k has a unique solution h = V P (f)k for all f in U and all k, and that the family of inverses V P : U × G → F is a smooth tame map. Then P is locally invertible, and each local inverse P^{−1} is smooth tame map.

Theorem 5. Suppose DP is surjective with smooth tame family of right inverse V P. Then P is locally surjective. Moreover in a neighborhood of any point, P has a smooth tame right inverse.

From the Nash-Moser inverse function theorem, Hamilton deduced the following (cf.[3])

Theorem 6. (Nash-Moser-Hamilton) Let G be a tame Fréchet Lie group acting tamely on a tame Fréchet manifold F with A : G × F → F the action. For any f ∈ F there is a linear map

A′_{f} = D_{G}A(e, f) : T_{e}G → T_{f}F.

Suppose that F is connected, and for each f ∈ F the map A′_{f} is surjective with a tame linear right inverse. Then G acts transitively on F.

Let (as in the previous sections) R^{n} be the Euclidean space endowed with the usual norm ∥.∥; let 𝟋 denote the Schwartz space of all functions on R^{n} which are fast falling together with all derivatives. 𝟋 is a tame Fréchet space (see [3]). Denote by E the graded Fréchet space of vectors fields with components in 𝟋 and by j_{0}^{1}E the closed subspace of E of vector fields which are 1-time flat at the origin 0 with the induced gradings. j_{0}^{1}E is a tame Fréchet space. Denote by G the group of diffeomorphisms f on R^{n} such that f − id ∈ j_{0}^{1}E ( 1-flat diffeomorphisms ). The group G modeled on the tame Fréchet space j_{0}^{1}E, is a tame Lie Fréchet group.

Canonically G acts on j_{0}^{1}E by the adjoint action

A : G × j_{0}^{1}E → j_{0}^{1}E A(f, X) = f_{∗}X = (Df.X)of^{−1}.

Let X_{o} be the infinitesimal dilatation, the affine space

𝟋 = {X_{o} + Z : Z ∈ j_{0}^{1}E}

is a tame Fréchet space and obviously 𝟋 is invariant by the Fréchet Lie group G. The tangent space T_{Xo}𝟋 at X_{o} of the space affine 𝟋 is identified to the Fréchet space j_{0}^{1}E. And also the tangent space T_{id}G to the group at the identity is identified to j_{0}^{1}E.

For any X_{0} ∈ j_{0}^{1}E the derivative of the orbital projection f → f_{∗}X_{0} writes as

A′_{X_{0}} = L (X_{0}) : j_{0}^{1}E → j_{0}^{1}E with L(X_{0})Y = ad_{Y} X_{0} = [Y, X_{0}] .

We are going to apply the Nash-Moser-Hamilton theorem to the above action; to reach this aim, we state.

Let ϕ_{t} = exp(tX_{0}), ψ_{t} = exp t(X_{o} + Z) with X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be the dilatation which all α_{i} < 0 and Z be a perturbation of X_{o} such that the perturbed dilatation Y_{o} = X_{o} + Z is admissible. Put f_{t} = ϕ_{−t}oψ_{t}.

Lemma 4. f_{t}−id is globally bounded to infinite order uniformly in t > 0.

Proof. For any multi-indices β = (β_{1}, ..., β_{n}) and any unit vector v = (v_{1}, ..., v_{n}), we have

D^{β}f_{t}(x)v^{β} = Dϕ_{−t}(ψ_{t}(x))D^{β}ψ_{t}(x)v^{β}

so

D^{β}f_{t}(x)′v^{β} = −(Dϕ_{−t}.X)(ψ_{t}(x))D^{β}ψ_{t}(x)v^{β}

+ (Dϕ_{−t}.D^{β}(X + Z)v^{β})oψ_{t}(x) = (Dϕ_{−t}(D^{β}Zv^{β}))oψ_{t}(x).

Letting in mind the notation introduced in subsection 2.3, by integrating we get

∥D^{β}(f_{t} − id)∥ ≤ ∫_{0}^{t} exp(bs) ∥D^{β}Z(ψ_{s}(x))∥ ds.

As in section 1, we obtain

D^{β}Z(ψ_{s}(x))v^{β} = D^{β}Z(y)v^{β}

= ∑_{l_{1}=1}^{β_{1}} ... ∑_{l_{n}=1}^{β_{n}} D_{y}^{l}Z(y) ∏_{ι=1}^{n} ∑_{η_{1}^{i}+...+η_{l_{i}}^{i}=β_{i}} D^{η_{1}^{i}}ψ_{t}(x)v_{1}^{η_{1}^{i}}...D^{η_{l_{i}}^{i}}ψ_{t}(x)v_{l_{i}}^{η_{l_{i}}^{i}}

and by Lemma 3, we have with y = ψ_{t}(x)

∥D_{x}^{β}Z(y)v^{β}∥ ≤ ∥Z∥_{r} exp(−na′_{1}t)

where |β| ≤ r .

Consequently for any any positive integer r and t ∈ R^{∗+}, we obtain

∥f_{t} − id∥_{r} ≤ ∥Z∥_{r} ∫_{0}^{t} e^{(b−na_{1})t}dt

so, if n ≥ 2 , we have the conclusion.

In the case n = 1, we use the flatness of Z at the origin 0. □

3.3. Right inverse of the operatorL(X_{o}). Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} be an admissible infinitesimal dilatation and consider the following equation

Z = L(X_{o})Y = [Y, X_{o}]

in the tame space j_{0}^{1}E.

Using the general relations

(ϕ_{t})_{∗}ad(X_{o}) = −\frac{d}{dt}(ϕ_{t})_{∗} = \frac{d}{dt}(ϕ_{t})^{∗},

we obtain by integrating

(I − ϕ_{t∗}) Y = ∫_{0}^{t}(ϕ_{s})_{∗}Zds.

Suppose that all the coefficients α_{i} > 0. Since by Lemma2, I − ϕ_{t∗} is invertible, we have

Y (Z) = (I − ϕ_{t∗})^{−1} ∫_{0}^{t}(ϕ_{s})_{∗}Zds for any t ∈ R^{+}.

Putting t = 1, we get

Y (Z) = (I − ϕ_{∗})^{−1} ∫_{0}^{1}(ϕ_{s})_{∗}Zds.

Letting

A = ∫_{0}^{1}(ϕ_{s})_{∗}Zds

be the mean adjoint operator we obtain

L(X)^{−1} = (I − ϕ_{∗})^{−1} A.

3.4. Tame property of the inverse L(X_{o})^{−1}. Suppose that the infinitesimal dilatation X_{o} is admissible with positive coefficients.

By Lemma 2, the inverse map is given by

(L(X))^{−1} : j_{0}^{1}E → j_{0}^{1}E , (L(X))^{−1} Z = Y (Z) = (I − ϕ_{∗})^{−1} A

and, by the general estimates (8), (9), and (10), we get for any integer r ≥ 1

∥(I − ϕ_{∗})^{−1} A∥_{r} ≤ C_{1} (∥(I − ϕ_{∗})^{−1}∥_{r} A_{0} + ∥(I − ϕ_{∗})^{−1}∥_{0} ∥A∥_{r}) ,
∥(I − ϕ_{∗})^{−1}∥_{r} ≤ C_{2} (1 + ∥I − ϕ_{∗}∥_{r}) ,
∥ϕ_{s}^{−1}∥_{r} ≤ C_{r} (1 + ∥ϕ_{s}∥_{r})

and

∥(ϕ_{s})_{∗}Z∥_{r} ≤ C_{3} (∥ϕ_{s}∥_{1} ∥ϕ_{s}^{−1}∥_{r} + ∥ϕ_{s}^{−1}∥_{0} ∥ϕ_{s}∥_{r}) ∥Z∥_{r}^{ψ_{s}(K)}

so

∥Y (Z)∥_{r} ≤ C_{4} (1 + ∥Z∥_{r+n}) .

Let K be a compact neighborhood of the origin 0 ∈ R.

Theorem 7. Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be the dilatation vector field, and Y_{o} = X_{o} + Z be an admissible perturbed vector field of X_{o} defined on K and with the perturbation Z having small support. Let F be the affine space of vector fields of the form X_{o} + Z . Let G be the group of diffeomorphisms of K which are close to the identity and have small support. Then G acts transitively on the space 𝟋.

Proof. Since 𝟋 is a tame Fréchet space and G is a Lie Fréchet group, by the Nash-Moser-Hamilton Theorem, we deduce that the Lie Fréchet group G acts transitively on 𝟋, that means that for every vector fields X_{1} and X_{2} in the space 𝟋 there is a diffeomorphism f ∈ G so that f_{∗}X_{1} = X_{2}.

In particular there is f ∈ G such that f_{∗}X_{o} = X_{o} + Z . □

4. Invertibility of the map X → exp X

4.1. X−derivative of the exponential map. Let χ_{K} be the Lie algebra of smooth vectors fields on a compact neighborhood of the origin 0 ∈ R^{n}. Each vector field X in χ_{K} induces a global one parameter group t → exp(tX), for t ∈ R. Thus we have the map

θ : χ_{K} × R^{n} × R → R^{n}; (X, x, t) → exp(tX)(x).

Since X is smooth, the map ϕ is also smooth in (x, t). The X-derivative at X in the direction of Y has been computed in [5] as

D exp(X)Y = ∫_{0}^{1}(exp sX)_{∗}Y ds ∘ exp X.  (11)

Let ϕ = exp(X), we have the mean adjoint operator A on χ_{K}

AY = ∫_{0}^{1}(ϕ_{s})_{∗}Y ds

and (11) writes as

DϕY = AY oϕ

so

AY = DϕY oϕ^{−1}.

4.2. Right inverse to the exponential map. Let as in the previous j_{0}^{1}E be the Fréchet space of smooth vector fields with components in the Schwartz space 𝟋 and which are 1-time flat at the origin, j_{0}^{1}E is a tame Fréchet space. Denote by G be the group of diffeomorphisms f on R^{n} such that f − id ∈ j_{0}^{1}E. The group G modeled on the tame space j_{0}^{1}E is a tame Lie Fréchet group. Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be an admissible infinitesimal dilatation. We are going to apply the Hamilton-Nash-Moser theorem to the mapping

P : Z ∈ j_{0}^{1}E → exp(X_{o} + Z) exp(−X_{o}) ∈ G.

P is a smooth tame map from the Fréchet space j_{0}^{1}E into the tame manifold G. With Z fixed, set ϕ = exp(X_{o}), ψ = exp(X_{o} + Z) and g = ψoϕ^{−1}.

The derivative of P at Z is

D_{Z}P.Y = Dψ ∫_{0}^{1}(ψ_{t})_{∗}Y dto(ψoϕ^{−1}) ∈ T_{g}G

where ψ_{t} = exp t(X_{o} + Z) and T_{g}G the tangent space to G at g. Then (Z, Y) → D_{Z}P.Y is a smooth tame family of maps from j_{0}^{1}E × j_{0}^{1}E into T_{g}G. Now, we look for a smooth family of inverses, that is solution Y_{Z,W} of the equation

D_{Z}P Y = W og

with Z and W from j_{0}^{1}E .

We have([5])

Proposition 1. If I −ψ_{∗} or I −ψ^{∗} is invertible on j_{0}^{1}E then the solution is given by

Y = ad(X_{o} + Z) (I − ψ_{∗})^{−1} W

or

Y = ad(X_{o} + Z) (I − ψ^{∗})^{−1} W

respectively.

We are going to show that the series

(I − ψ_{∗})^{−1}Y (x) = ∑_{m≥0}(ψ_{m})_{∗}Y (x) = ∑_{m≥0}(ψ_{−m})^{∗}Y (x)

converges.

Lemma 5. Under the conditions of Lemma2, the series

(I − ψ_{∗})^{−1}Y (x) = ∑_{m≥0} (ψ_{m})_{∗} Y (x)

(resp. the series (I − ψ^{∗})^{−1}Y (x) = ∑_{m≥1}(ψ_{m})^{∗}Y (x) ) converges on the Fréchet space j_{0}^{1}E.

Proof. Putting Z(t) = ϕ_{t}^{∗}Y and using the estimates 8, 9 and 10, we get for any k ≥ 1,

∥D^{k}f_{t}^{∗}Z(t)∥ = ∥D^{k}(Df_{−t}Z)of_{t}∥

≤ C_{1}(∥f_{t}∥_{1}^{k−1} ∥f_{t}∥_{1} ∥Df_{−t}Z∥_{k} + ∥Df_{−t}Z∥_{1} ∥f_{t}∥_{k})
≤ C_{1} ∥f_{t}∥_{1}^{k−1} {C_{2} ∥f_{t}∥_{1} (∥f_{−t}∥_{k+1} ∥Z∥_{0} + ∥f_{−t}∥_{2} ∥Z∥_{k}
+ C_{3} ∥f_{t}∥_{k} ∥f_{−t}∥_{2} ∥Z∥_{0} + ∥f_{−t}∥_{1} ∥Z∥_{1}}

≤ C ∥f_{t}∥_{1}^{k−1} (∥f_{t}∥_{1} ∥f_{−t}∥_{k+1} + ∥f_{t}∥_{k} ∥f_{−t}∥_{2}) ∥Z∥_{k} .

By Lemma 4, all ∥f_{t}∥_{j} are independent of t and, for any Y ∈ j_{0}^{1}E , one obtains the estimate

∥(ψ_{t})_{∗}Y ∥_{k} = ∥f_{t}_{∗}(ϕ_{t∗}Y)∥_{k} ≤ C ∥Z(t)∥_{k} ≤ C ∥ϕ_{t∗}Y ∥_{k} .

Since, by Lemma 2, the series ∑_{m≥1} ∥(ϕ_{t})_{∗}Y ∥_{k} tamely converge, so do the one for ψ_{t}. □

Lemma 6. The map (Z, Y) ∈ j_{0}^{1}E × j_{0}^{1}E → (I − (ψ_{t})_{∗})^{−1}Y ∈ j_{0}^{1}E is smooth and gives a tame family of inverses.

Proof. Since, by Lemma 5, (I − ψ_{t})_{∗}) is tamely invertible on j_{0}^{1}E and the map Z → exp(X_{o} + Z) is smooth and tame so do the map given in Lemma 6. □

We have checked the hypothesis of the Nash-Moser theorem, now we get the main result of this section

Theorem 8. Let X_{o} = ∑_{i=1}^{n} α_{i}x_{i} \frac{∂}{∂x_{i}} be the dilatation vector field and 𝟋 the affine space of admissible vector fields Y_{o} = X_{o} + Z of X_{o} defined on R^{n}and with Z of small support. Then for every diffeomorphism f on R^{n} which is 1-time flat, close to the identity and of small support, there exists a vector field Z such that X_{o} + Z ∈ 𝟋, fulfilling

f = exp(X_{o} + Z)o exp(−X_{o}).

In [1], using the Steinberg linearization theorem, the first author obtained the following result concerning germs at the origin 0 similar to Theorem 8.

Theorem 9. Let f be a germ of diffeomorphism at the origin 0 of R^{n} which is 1-time flat at 0 and A be a germ of a linear vector field given in coordinates by Ax = (α_{1}x_{1}, ..., α_{n}x_{n}) where α_{i} satisfy the additive Steinberg condition α_{i} ≠ ∑_{i=1}^{n} m_{i}α_{i} for all nonnegative integers m_{i} satisfying 2 ≤ m_{1} +...+m_{n}. Then there exists a germ of vector field X on R^{n} which is 1-time flat at the origin 0 such that f = exp(−A)o exp(A + Z).

References

[1] Benalili, M. Fibrés principaux Γ−naturels. Demonst. math. VolXXV, 4 (1992), 906-925.

[2] Grabowski, J. Derivative of the exponential mapping for infinite dimensional Lie groups. Annals Global.Geom. 11 (1993), 213-220

[3] Hamilton, R.S. The inverse function theorem of Nash and Moser . Bull.Amer.Math.Soc.7 (1982), 65-222.

[4] Lansari A. Stabilité globale d’ordre supérieur (article in preparation )

[5] Zajtz, A. Calculus of flows on convenient manifolds Archivum mathematicum Tomus 32 1996 .

University Aboubekr-Belkaïd Faculty of Sciences, BP119. Tlemcen Algerie. E-mail address: m benalili@yahoo.fr E-mail address: a lansari@yahoo.fr

Received July 6, 2005